Consider a long infinite current carrying wire in N-S direction.
Consider a charge +q at a distance of r from wire.
Clearly, Lorentz force on charge is 0.
But, their is a relative motion between charge and current in wire.
Can't this cause Lorentz force?
Magnetic field will not apply any force on the charge if the charge does not have a velocity (or a component of velocity)that's perpendicular to its own direction. A magnet is basically a collection of current carrying loops. Since current is basically movement of electrons, electroSTATIC force has nothing to do with it. A current carrying loop will attract another current carrying loop. Not any random charge that too at rest.
the length contraction will happen but we must consider the fact that this is happening because a frame is moving at the same speed as the current making charge particles. However at the same time we can easily observe that the wire in which current is there is moving backwards at the same speed or the positive charged atoms or regions are moving backward so thereby again constituting the same current which in turn gives the same magnetic field that's observed by the rest frame which might feel length contraction..so in both frame of references there shouldn't be any electric field.
Nowhere in the statement of the question do you explicitly state that the current carrying wire is uncharged, but since that is the usual assumption I will assume that is what was intended.
Then the analysis of the force on the charge is clear. $F= q(E+v\times B)$. The first term is 0 because the wire is uncharged, and the second term is also 0 because the charge is stationary. Therefore the Lorentz force is unambiguously 0.
The exact motion of the moving charge carriers is not directly relevant (a large number of slow charges makes the same current as a small number of fast charges). The relevant question is whether or not the wire is charged in the rest frame and what current it carries.
